Solving Equations

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Solving Equations

  • Finding all solutions of an equation is called solving the equation.
  • To solve an equation, isolate the variable.
  • A variable is isolated when it appears by itself on one side of an equation.
  • Isolate a variable by using inverse operations, which undo operations on the variable.
  • When you perform the same operation on both sides of an equation, the two sides remain equal. (Properties of Equality)
Illustrating how to solve an equation using inverse operations

Solving Equations

Properties of Equality

  • Properties of Equality

    • You can perform the same operation on both sides of an equation.
Properties of Equality explained

Properties of Equality


isolate を分離する

remain {引き続き}~のままである

perform {楽しませるため}(を)演じる;



In solving equations, the concept of ‘isolating’ the variable x is introduced. The method of isolation involves the following repetition:

  1. Remove the obstacles with inverse operations.
  2. Perform the same operation on the opposite side in order to keep the equality.

Mastering the properties of equality is extremely important!
Including similar techniques, namely adding (or multiplying) something convenient for yourself and then subtracting (or multiplying by the reciprocal) for consistency, these methods will appear everywhere in the future.
For example:

  • Solving simultaneous equations
  • Completing the square in quadratic functions
  • Composition of trigonometric functions (in high school)

Perhaps to demonstrate this importance, many textbooks describe all four arithmetic operations of the properties of equality — addition, subtraction, multiplication, and division — in both text and formulas.
Textbooks in Japan, too!


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